Given $ m \angle RPS = 8x - 41$, $ m \angle QPR = 6x + 75$, and $ m \angle QPS = 146$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {6x + 75} + {8x - 41} = {146}$ Combine like terms: $ 14x + 34 = 146$ Subtract $34$ from both sides: $ 14x = 112$ Divide both sides by $14$ to find $x$ $ x = 8$ Substitute $8$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 6({8}) + 75$ Simplify: $ {m\angle QPR = 48 + 75}$ So ${m\angle QPR = 123}$.